Abstract
We introduce the concept of ψ-firmly nonexpansive mapping, which includes a firmly nonexpansive mapping as a special case in a uniformly convex Banach space. It is shown that every bounded closed convex subset of a reflexive Banach space has the fixed point property for ψ-firmly nonexpansive mappings, an important subclass of nonexpansive mappings. Furthermore, Picard iteration of this class of mappings weakly converges to a fixed point. MSC:47H06, 47J05, 47J25, 47H10, 47H17.
Highlights
Throughout this paper, a Banach space E will be over the real scalar field
Let F(T) = {x ∈ E : Tx = x}, the set of all fixed points for a mapping T and N denote the set of all positive integer
We introduce the concept of ψ -firmly nonexpansive mapping which includes the firmly type nonexpansive mapping as a special case (ψ(t) = kt )
Summary
Throughout this paper, a Banach space E will be over the real scalar field. We denote its norm by · and its dual space by E∗. The famous question whether a Banach space has the fixed point property (WFPP) had remained open for a long time [ , ] It has been answered in the negative by Sadovski [ ] and Alspach [ ] who constructed the following examples, respectively. We will study fixed point properties of ψ -firmly nonexpansive mapping, an important subclass of nonexpansive mappings, on weakly compact convex subsets of a Banach space. I is an identity operator and T is a nonexpansive self-mapping defined on a nonempty bounded closed convex subset K of E, for each x ∈ K , Picard iteration {Snx} weakly converges to a fixed point of T; Bruck [ , ] proved that for each x. We show that in a uniformly convex space, ψ -firmly nonexpansive mapping includes a firmly nonexpansive mapping and the resolvent of an accretive operator as a special case
Sign-in/Register to view highlights & summary Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
